Question: $ -1.\overline{6} \div 0.\overline{83} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -16.6667...\\ x &= -1.6667...\end{align*} $ $\begin{align*} 9x &= -15 \\ x &= -\dfrac{15}{9}\end{align*} $ $\begin{align*} 100y &= 83.8383...\\ y &= 0.8383...\end{align*} $ $\begin{align*} 99y &= 83 \\ y &= \dfrac{83}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{15}{9} \div \dfrac{83}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{15}{9} \times \dfrac{99}{83} = {?} $ $ \phantom{-\dfrac{15}{9} \times \dfrac{83}{99}} = \dfrac{-15 \times 99}{9 \times 83} $ $ \phantom{-\dfrac{15}{9} \times \dfrac{83}{99}} = \dfrac{-15 \times \cancel{99}11} {\cancel{9} \times 83} $ $ \phantom{-\dfrac{15}{9} \times \dfrac{83}{99}} = -\dfrac{165}{83} $